Advanced book on Mathematics Olympiad

772 Combinatorics and Probability We count these points using the inclusion–exclusion principle. The first coordinate of the cur ...

Combinatorics and Probability 773 908.Denote byP (n)the probability that a bag containingndistinct pairs of tiles will be emptie ...

774 Combinatorics and Probability 911.Consider the dual cube to the octahedron. The verticesA,B,C,D,E,F,G, Hof this cube are the ...

Combinatorics and Probability 775 912.The total number of permutations is of coursen!. We will count instead the number of permu ...

776 Combinatorics and Probability which follows from the deragements formula (see Section 6.2.4.). The probability that exactlyk ...

Combinatorics and Probability 777 E(X^2 i)=E(Xi)= 1 n and E(XiXj)= 1 · 1 ·P(Xi= 1 ,Xj= 1 )= 1 n(n− 1 ) . HenceE(X^2 )= 1 + 1 =2. ...

778 Combinatorics and Probability P (B/A)= P(B) P (A) ·P(A/B). For our problemAis the event that the mammogram is positive andBt ...

Combinatorics and Probability 779 P(min(x◦,y◦)= 70 ◦)=P(x◦= 70 ◦)+P(y◦= 70 ◦)−P(max(x◦,y◦)= 70 ◦) =a+b−c. (29th W.L. Putnam Math ...

780 Combinatorics and Probability Second solution: Using the Poisson scheme (p 1 x+ 1 −p 1 )(p 2 x+ 1 −p 2 )(p 3 x+ 1 −p 3 )= 2 ...

Combinatorics and Probability 781 1 − 95 100 · 94 99 · 93 98 · 92 97 · 91 96 ≈ 0. 230. 923.We apply Bayes’ formula. LetBbe the e ...

782 Combinatorics and Probability P(k)=pP (k+ 1 )+qP(k− 1 )+rP (k). Taking into account thatp+q+r=1, we obtain the recurrence re ...

Combinatorics and Probability 783 The favorable cases consist of the region Df= { (x, y)∈D|x+y≤ 1 ,xy≤ 2 9 } . This is the set o ...

784 Combinatorics and Probability D={(x, y)| 0 ≤x≤ 1 , 0 ≤y≤ 1 }. In order for the two people to meet, their arrival time must l ...

Combinatorics and Probability 785 Let us first measure the volume of the configurations(P 1 ,P 2 ,P 3 )such that the arc  P 1 P ...

786 Combinatorics and Probability 933.The pair(p, q)is chosen randomly from the three-dimensional domainC×intC, which has a tota ...

Combinatorics and Probability 787 x θ Figure 113 935.First solution: We will prove that the probability is 1− 1235 π 2. To this ...

788 Combinatorics and Probability similarly forr 2. The distribution ofθis uniform on[ 0 ,π]. These three distributions are inde ...

Combinatorics and Probability 789 = 2 E′(A(OP R))− 2 3 E′(χ A(OP R))= 29 36 π . Finally, note that the case in whichP′,Q′,R′lie ...

790 Combinatorics and Probability the area we are seeking (after doubling) is 2 1 +r^2 − 2 h^2 √ r^2 −h^2 . Dividing byπ, then i ...

Index of Notation N the set of positive integers 1, 2 , 3 ,... Z the set of integers Q the set of rational numbers R the set of ...